where
LR = maximum rent that a particular agricultural use can pay at a particular point, in dollars/acre
Q = output (a measure of intensity of use), in tons/acre
p = market price (a measure of value of the crop), in dollars/ton
c = production cost, in dollars/ton
f = transport cost, in dollars/ton-mile
k = distance from the market center to the point in question, in miles.

Recall that this equation yields a landscape of concentric circles of different agricultural land uses, centered on the market, because different land uses have different outputs per acre and different transport costs.

Well, we can use this same logic to conceptualize the pattern of urban land uses.  At the urban scale, even small users of land matter:  houses, factories, train stations, and office buildings that appear as “dots” in the market center of an agricultural tableau are noticeable users of land at the urban scale.

Each land use has a “profitability” (r - c) — the amount of revenue generated per square foot minus the costs of generating that revenue:

• for offices, this is the rent that office functions will pay per square foot, minus the cost of maintaining the office;
• for manufacturing or warehousing facilities, this is the operating revenue generated per square foot of floor space, minus the operating costs of production;
• for retailing, this is the sales per square foot minus the costs of supplies and labor;
• for housing, this is the rent per square foot, minus the costs of maintaining the building and grounds.

• Offices, stores, and housing units can be stacked atop one another in tall buildings, packed densely alongside one another in low buildings, or can be widely dispersed, with space around each building.
• Manufacturing and warehousing have smaller ranges of economical densities.

• there are daily journeys-to-work, journeys-to-school, and regular journeys-to-shop
• goods are moved from other cities through seaports, airports, rail terminals, and along major highways

To the extent that the center is the point of maximum accessibility, it the point where transport costs are lowest.  So we can suggest that people will pay higher economic rent to locate nearer the city center, where transport costs are lowest to the combination of all other points.

What is this equation saying?
(r - c) is the profit to be made per sq.ft. of rented space:  higher for high-volume department stores and for exclusive office space for corporate headquarters, lower for inexpensive housing or land-hungry manufacturing.
D (r - c) is the profit to be made per acre of land:  this gets pretty high even for inexpensive housing, if the density is great enough.
Dtd_ic is the cost to transport the people (and goods) that have to get to and from the site every day:  the denser the land use and the more people who use it per square foot of space, the more costly it is to be far from the point of maximum accessibility to everyone’s home and workplace.

The equation is saying that the portion of the economic return from an urban activity which can be attributed to the location of the activity equals the difference between (a) the profit to be made per acre of land, before transport costs and (b) the costs associated with moving goods and people to and from the location.

We can relate this equation to our linear rent gradients in the form Y = a - bX:  D (r - c) is the y-intercept (the rent at the point of maximum accessibility) and Dt is the slope.

If we assume that different urban land uses have different densities, we can understand a pattern of concentric zones around the point or points of maximum accessibility.  Hanink's Figure 2.9 and Stutz & deSouza's Figures 6.5 and 6.6 show this when there is one point of maximum accessibility;  we can also show a more complicated pattern when there are several points of great accessibility, as where a circumferential highway (the Beltway around Washington, D.C., or Route 128 and Route 495 around Boston) intersects the radial highways leading into the center of town.

Note that this framework has the same assumptions (follow this link)  and is built in the same general way (follow this link) as the agricultural land use model.  This general approach, its strengths (simplicity), and its weaknesses (its strong assumptions) should become very familiar to you by the end of this course.
 

COMPLICATIONS  AND  TRANSITIONS

The resultant urban land uses are complicated by the dynamic nature of urban development:  as the city grows, each ring gets larger.

[illustrate the rings expanding, as the prices at the center rise with greater overall population]

 But there are already buildings and infrastructure from the earlier land uses.  So we get “zones of transition,” where former warehouses or manufacturing lofts are used for a mix of functions, until the land becomes so valuable that they are demolished and the land used for higher-density office or residential functions.

The pattern is also disturbed by linear or radial infrastructure that is in place as a city grows:  major rail lines that remain industrial over time, or landscaped parkland that is desired by the most expensive housing.  This leads to a sector pattern of urban land uses.

Note how we just “played with” this model:  we asked what would happen over time if the total size of the city increased.  The answer, in this simple model, is that the rents rise throughout the area:  the zones increase in size, and the spatial margin of the city expands.  To use the algebraic terms, r increases, which drives up the y-intercept of each land-use rent gradient, without affecting the slopes of the gradients.

• See Hanink, pp. 106-109, for a discussion (and diagrams) of the implications for urban sprawl and the shrinkage of wild land.
• Have you heard of the Urban Growth Boundary around the urbanized portions of King County, Washington (Seattle)?  See two recent Seattle Times articles (second article) on the difficulty of managing urban sprawl.
• See the Brookings Institution policy brief on urban form.
• See Hanink's Case 4.2, pp. 131-132.  What impact do you think the preservation of farm land will have on urban land values?  What impact do you think it will have on urban development density?  What impact do you think it will have on the lifestyles and day-to-day activities of urban residents?  Of "close-in" farm residents?

If gasoline prices were to rise substantially (say, by a factor of three), the sloped lines would get steeper, the zones and the spatial extent of the city would get smaller, rents would increase in more accessible locations, and densities would increase.  To use the algebraic terms, t would increase, which increases the slope of each land-use gradient, while the y-intercepts remain the same.

What happens if the dominant density of a given land use changes?  During the middle of the 20th century, manufacturing technology changed manufacturing from a fairly dense land use to a fairly land-hungry land use.  To use the algebraic terms, D has fallen for this particular land use, which decreases the y-intercept and the slope for this land use.  Correspondingly, manufacturing has shifted from inner to outer rings of most metropolitan areas.

POLITICAL  REALITIES

Land-use zoning
The actual patterns of land use in a North American city (except Houston) reflect the decisions of local governments regarding land-use zoning.  These decisions generally follow the market, however, because

(a) their purpose is merely to separate incompatible uses from being adjacent, not to change the form of the city and
(b) it’s usually pretty easy to get a variance from a zoned land use.

Balkanization of the metropolis
Most large metropolitan areas are actually divided into many separate municipalities, each one collecting its own property taxes and supporting its own schools, fire departments, etc.  As the metropolitan area expands its spatial extent, a smaller and smaller proportion of it is within the central city.  The central city then has to support big-city functions (major library, large police force for all the tourists and in-commuting workers, extra educational expenses for populations that need additional educational assistance) with a shrinking proportion of the region’s tax base:  taxes per household increase, which accelerates the process of suburbanization.

Informal housing
Most housing in U.S. metropolitan areas is built professionally and according to building codes.  In Latin America, however, poor households may build housing themselves on land that they don’t formally own or rent.  These informal housing settlements are generally on the edge of town — so the pattern of housing is the opposite of most North American cities, where most wealthy people live in land-use-controlled suburbs and most poor, urban residents live in the central city.